In this section we prove our main result.

5.1. Theorem.*The* Cat*-functor* U :ClMulticat^{u} →ClCat *is a* Cat*-equivalence.*

We have to prove that U is bijective on 1-morphisms and 2-morphisms, and that it is essentially surjective; the latter means that for each closed category V there is a closed multicategory with a unit object such that its underlying closed category is isomorphic (as a closed category) to V.

5.2. The surjectivity of U on 1-morphismsLetCand Dbe closed multicategories
with unit objects. Denote their underlying closed categories by the same symbols. Let
Φ = (φ,φ, φˆ ^{0}) : C → D be a closed functor. We are going to define a multifunctor

F : C→ Dwhose underlying closed functor is Φ. Define F X =φX, for each X ∈ ObC.
For each Y ∈ObC, the map F_{;Y} :C(;Y)→D(;φY) is defined via the diagram

C(;Y) D(;φY)

C(1;Y) D(φ1;φY) D(1;φY)

F_{;Y}

C(u;1) ≀ ≀ D(u;1)

φ D(φ^{0};1)

Recall that for a morphism f : ()→Y we denote by f :1→Y a unique morphism such that u·f =f. Then the commutativity in the above diagram means that

F f =

()−→^{u} 1 ^{φ}

−→0 φ1−−→^{φ(f)} φY

, (5.1)

for each f : ()→Y. For n≥1 and X1, . . . , Xn, Y ∈ObC, the map
FX_{1},...,Xn;Y :C(X1, . . . , Xn;Y)→D(φX1, . . . , φXn;φY)
is defined inductively by requesting the commutativity in the diagram

C(X2, . . . , Xn;C(X1;Y)) D(φX2, . . . , φXn;φC(X1;Y))

D(φX2, . . . , φXn;D(φX1;φY))

C(X1, . . . , Xn;Y) D(φX1, . . . , φXn;φY)

F_{X}_{2,...,Xn}_{;}C(X1;Y)

ϕ^{C} ≀

D(1; ˆφ)

ϕ^{D}

≀ FX1,...,Xn;Y

(5.2)

5.3. Lemma.*The following diagram commutes*

C(;C(X;Y)) D(;φC(X;Y)) D(;D(φX;φY))

C(X;Y) D(φX;φY)

F_{;}C(X;Y) D(; ˆφ)

ϕ^{C} ≀ ≀ ϕ^{D}

φ

*In particular,* FX;Y =φX,Y :C(X;Y)→D(φX;φY).

Proof.Equivalently, the exterior of the diagram

C(;C(X;Y)) D(;φC(X;Y)) D(;D(φX;φY))

C(1;C(X;Y)) D(φ1;φC(X;Y)) D(1;φC(X;Y)) D(1;D(φX;φY))

C(X;Y) D(φX;φY)

F_{;}C(X;Y) D(; ˆφ)

C(u;1)

≀ D(u;1) ≀ D(u;1) ≀

φ D(φ^{0};1) D(1; ˆφ)

γ γ

φ

(ϕ^{C})^{−1} (ϕ^{D})^{−1}

commutes. The upper pentagon is the definition of F;C(X;Y). The bottom hexagon com-mutes. Indeed, taking f ∈C(X;Y) and tracing it along the left-top path yields

φ^{0}·φ(jX)·φC(1;f)·φˆ=φ^{0}·φ(jX)·φˆ·D(1;φ(f)) (naturality of ˆφ)

=jφX ·D(1;φ(f)), (axiom CF1) which is precisely the image of f along the bottom-right path.

5.4. Lemma.*For each* f : () →Y *and* Z ∈ObC*, the diagram*
φC(Y;Z) φC(;Z) φZ

D(φY;φZ) D(;φZ) φZ

φC(f;1)

φˆ

D(F f;1)

*commutes.*

Proof.By definition,

F f =

()−→^{u} 1−→^{φ}^{0} φ1−−→^{φ(f)} φY
.
The diagram

φC(Y;Z) φC(1;Z) φC(;Z) φZ

D(φY;φZ) D(φ1;φZ) D(;φZ) φZ

φC(f;1) φC(u;1)

φˆ φˆ

D(φ(f);1) D(u·φ^{0};1)
φC(f;1)

D(F f;1)

commutes. Indeed, the left square commutes by the naturality of ˆφ, while the commuta-tivity of the right square is a consequence of the axiom CF2, see (4.1).

With the notation of Lemma 3.14, we can rewrite the commutativity condition in diagram (5.2) as a recursive formula for the multigraph morphism F:

F f =ϕ^{D}(F((ϕ^{C})^{−1}(f))·φ) =ˆ ϕ^{D}(Fhfi ·φ),ˆ
for each f :X1, . . . , Xn →Y with n≥1, or equivalently

hF fi=

φX_{2}, . . . , φXn
Fhfi

−−→φC(X1;Y)−→^{φ}^{ˆ} D(φX1;φY)

. (5.3)

5.5. Lemma.*For each* X, Y, Z ∈ObC*, the diagram*

φC(X;Y), φC(Y;Z) φC(X;Z)

D(φX;φY),D(φY;φZ) D(φX;φZ)

F µC

φ,ˆφˆ φˆ

µD

*commutes.*

Proof.It suffices to prove the equation

hF µC·φiˆ =h( ˆφ,φ)ˆ ·µDi.

By Lemma 3.14,(c), the left hand side is equal to

φC(Y;Z)−−−→^{hF µ}^{C}^{i} D(φC(X;Y);φC(X;Z))−−−→^{D}^{(1; ˆ}^{φ)} D(φC(X;Y);D(φX;φZ)),
while the right hand side is equal to

φC(Y;Z)−→^{φ}^{ˆ} D(φY;φZ)−−→^{hµ}^{D}^{i} D(D(φX;φY);D(φY;φZ))−−−→^{D}^{( ˆ}^{φ;1)} D(φC(X;Y);D(φX;φZ))
by Lemma 3.14,(b). Note that hµ^{D}i= (ϕ^{D})^{−1}(µ^{D}) =L^{φX}. Furthermore, by (5.3),

hF µCi=

φC(Y;Z)−−−→^{φhµ}^{C}^{i} φC(C(X;Y);C(X;Z))−→^{φ}^{ˆ} D(φC(X;Y);φC(X;Z))

=

φC(Y;Z)−−→^{φL}^{X} φC(C(X;Y);C(X;Z))−→^{φ}^{ˆ} D(φC(X;Y);φC(X;Z))
,
therefore the equation in question is simply the axiom CF3.

5.6. Proposition. *The multigraph morphism* F : C → D *is a multifunctor, and its*
*underlying closed functor is* Φ.

Proof.Trivially, F preserves identities since so does φ. Let us prove that F preserves composition. The proof is in three steps.

5.7. Lemma.F *preserves composition of the form* X1, . . . , Xk

−f

→Y −→^{g} Z.

Proof. The proof is by induction on k. There is nothing to prove in the case k = 1.

Suppose that k = 0 and we are given composable morphisms
()−→^{f} X −→^{g} Y.

Then since u·f g = f · g = (u ·f) ·g = u·(f ·g), it follows that f·g = f ·g. By formula (5.1),

F(f ·g) =u·φ^{0}·φ(f·g) = u·φ^{0}·φ(f ·g) =u·φ^{0}·φ(f)·φ(g) =F f·F g.

Suppose that k >1. Then

hF(f·g)i=Fhf·gi ·φˆ (formula (5.3))

=F(hfi ·C(1;g))·φˆ (Lemma 3.14,(c))

=Fhfi ·φC(1;g)·φˆ (induction hypothesis)

=Fhfi ·φˆ·D(1;φ(g)) (naturality of ˆφ)

=hF fi ·D(1;F g) (formula (5.3))

=hF f·F gi, (Lemma 3.14,(c)) and induction goes through.

5.8. Lemma.F *preserves composition of the form*

X_{1}^{1}, . . . , X_{1}^{k}^{1}, X_{2}^{1}, . . . , X_{2}^{k}^{2} −−−→^{f}^{1}^{,f}^{2} Y_{1}, Y_{2} −→^{g} Z

*.*

Proof.The proof is by induction on k_{1}. If k_{1} = 0, then by Lemma 3.14,(a),
(f1, f2)·g =

X_{2}^{1}, . . . , X_{2}^{k}^{2} −^{f}→^{2} Y2

−→hgi C(Y1;Z)−−−−→^{C}^{(f}^{1}^{;1)} C(;Z) =Z
,
therefore

F((f1, f2)·g) =F f2·φhgi ·φC(f1; 1) (Lemma 5.7)

=F f_{2}·φhgi ·φˆ·D(φ(f1); 1) (Lemma 5.4)

=F f2· hF gi ·D(F f1; 1) (formula (5.3))

= (F f1, F f2)·F g. (Lemma 3.14,(a))
If k_{1} = 1, then by Lemma 3.14,(b),

h(f1, f2)·gi=

X_{2}^{1}, . . . , X_{2}^{k}^{2} −^{f}→^{2} Y2

−→hgi C(Y1;Z)−−−−→^{C}^{(f}^{1}^{;1)} C(X_{1}^{1};Z)
,

therefore

hF((f1, f_{2})·g)i=Fh(f1, f_{2})·gi ·φˆ (formula (5.3))

=F f_{2} ·φhgi ·φC(f_{1}; 1)·φˆ (Lemma 5.7)

=F f2 ·φhgi ·φˆ·D(φ(f1); 1) (naturality of ˆφ)

=F f2 · hF gi ·D(F f1; 1) (formula (5.3))

=h(F f1, F f2)·F gi, (Lemma 3.14,(b))

and henceF((f1, f2)·g) = (F f1, F f2)·F g. Suppose thatk1 >1. Then by Lemma 3.14,(c)
h(f1, f_{2})·gi is equal to the composite

X_{1}^{2}, . . . , X_{1}^{k}^{1}, X_{2}^{1}, . . . , X_{2}^{k}^{2} −−−−→^{hf}^{1}^{i,f}^{2} C(X_{1}^{1};Y1), Y2
1,hgi

−−→C(X_{1}^{1};Y1),C(Y1;Z)−→^{µ}^{C} C(X_{1}^{1};Z),
therefore

hF((f1, f2)·g)i=Fh(f1, f2)·gi ·φˆ (formula (5.3))

= (Fhf1i, F f2)·F((1,hgi)µ^{C})·φˆ (induction hypothesis)

= (Fhf_{1}i, F f_{2})·(1, Fhgi)·F µC·φˆ (casek_{1} = 1)

= (Fhf1i, F f2)·(1, Fhgi)·( ˆφ,φ)ˆ ·µD (Lemma 5.5)

= (Fhf1i ·φ, F fˆ _{2})·(1, Fhgi ·φ)ˆ ·µD

= (hF f1i, F f2)·(1,hF gi)·µD (formula (5.3))

=h(F f1, F f2)·F gi, (Lemma 3.14,(c))
hence F((f1, f_{2})·g) = (F f1, F f_{2})·F g, and the lemma is proven.

5.9. Lemma.F *preserves composition of the form*

X_{1}^{1}, . . . , X_{1}^{k}^{1}, . . . , X_{n}^{1}, . . . , X_{n}^{k}^{n} −−−−→^{f}^{1}^{,...,f}^{n} Y_{1}, . . . , Yn

−g

→Z. (5.4)

Proof.The proof is by induction on n, and for a fixed n by induction on k_{1}. We have
worked out the cases n = 1 andn = 2 explicitly in Lemmas 5.7 and 5.8. Assume that F
preserves an arbitrary composition of the form

U_{1}^{1}, . . . , U_{1}^{l}^{1}, . . . , U_{n−1}^{1} , . . . , U_{n−1}^{l}^{n−1} −−−−−−→^{p}^{1}^{,...,p}^{n}^{−1} V1, . . . , Vn−1

−q

→W,

and suppose we are given composite (5.4). We do induction on k_{1}. If k_{1} = 0, then by
Lemma 3.14,(a) (f_{1}, . . . , fn)·g is equal to the composite

X_{2}^{1}, . . . , X_{2}^{k}^{2}, . . . , X_{n}^{1}, . . . , X_{n}^{k}^{n} −−−−→^{f}^{2}^{,...,f}^{n} Y_{2}, . . . , Yn

−→hgi C(Y1;Z)−−−−→^{C}^{(f}^{1}^{;1)} C(;Z) =Z,

therefore

F((f_{1}, . . . , fn)·g) = (F f_{2}, . . . , F fn)·F(hgi ·C(f_{1}; 1)) (induction hypothesis)

= (F f_{2}, . . . , F fn)·(Fhgi ·φC(f_{1}; 1)) (Lemma 5.7)

= (F f2, . . . , F fn)·(Fhgi ·φˆ·D(φ(f1); 1)) (Lemma 5.4)

= (F f2, . . . , F fn)·(hF gi ·D(F f1; 1)) (formula (5.3))

= (F f1, . . . , F fn)·F g. (Lemma 3.14,(a))

Suppose thatk_{1} = 1. Then by Lemma 3.14,(b) h(f1, . . . , fn)·gi is equal to the composite
X_{2}^{1}, . . . , X_{2}^{k}^{2}, . . . , X_{n}^{1}, . . . , X_{n}^{k}^{n} −−−−→^{f}^{2}^{,...,f}^{n} Y_{2}, . . . , Yn

−→hgi C(Y1;Z)−−−−→^{C}^{(f}^{1}^{;1)} C(X_{1}^{1};Z),
therefore

hF((f1, . . . , fn)·g)i=Fh(f1, . . . , fn)·gi ·φˆ (formula (5.3))

= (F f2, . . . , F fn)·F(hgi ·C(f1; 1))·φˆ (induction hypothesis)

= (F f2, . . . , F fn)·Fhgi ·φC(f1; 1)·φˆ (Lemma 5.7)

= (F f2, . . . , F fn)·Fhgi ·φˆ·D(φ(f1); 1) (naturality of ˆφ)

= (F f2, . . . , F fn)· hF gi ·D(F f1; 1) (formula (5.3))

=h(F f_{1}, . . . , F fn)·F gi, (Lemma 3.14,(b))

and hence F((f1, . . . , fn) · g) = (F f1, . . . , F fn) · F g. Suppose that k_{1} > 1, then by
Lemma 3.14,(c) h(f1, . . . , fn)·gi is equal to the composite

X_{1}^{2}, . . . , X_{1}^{k}^{1}, X_{2}^{1}, . . . , X_{2}^{k}^{2}, . . . , X_{n}^{1}, . . . , X_{n}^{k}^{n} −−−−−−−→^{hf}^{1}^{i,f}^{2}^{,...,f}^{n} C(X_{1}^{1};Y1), Y2, . . . , Yn
1,hgi

−−−−−−−→C(X_{1}^{1};Y_{1}),C(Y1;Z)

µC

−−−−−−−→C(X_{1}^{1};Z),
therefore

hF((f_{1}, . . . , fn)·g)i=Fh(f_{1}, . . . , fn)·gi ·φˆ (formula (5.3))

= (Fhf1i, F f2, . . . , F fn)·F((1,hgi)µC)·φˆ (induction hypothesis)

= (Fhf1i, F f2, . . . , F f_{n})·(1, F[g])·F µC·φˆ (Lemma 5.8)

= (Fhf_{1}i, F f_{2}, . . . , F fn)·(1, Fhgi)·( ˆφ,φ)ˆ ·µD (Lemma 5.5)

= (Fhf1i ·φ, F fˆ 2, . . . , F fn)·(1, Fhgi ·φ)ˆ ·µD

= (hF f1i, F f2, . . . , F fn)·(1,hF gi)·µD (formula (5.3))

=h(F f_{1}, . . . , F fn)·F gi, (Lemma 3.14,(c))
hence F((f1, . . . , fn)·g) = (F f1, . . . , F fn)·F g, and induction goes through.

Thus we have proven that F : C → D is a multifunctor. By construction, its
un-derlying functor is φ. Furthermore, the closing transformation F_{X}_{;Y} coincides with
φˆX,Y : φC(X;Y) → D(φX;φY). Indeed, we first observe that F_{X,Y} = hF ev^{C}i, where
closed functor is Φ. The proposition is proven.

5.10. The injectivity of U on 1-morphisms The following proposition shows that the Cat-functor U is injective on 1-morphisms.

5.11. Proposition. *Let* F, G : C → D *be multifunctors between closed multicategories*
*with unit objects. Suppose that* F *and* G *induce the same closed functor* Φ = (φ,φ, φˆ ^{0})
*between the underlying closed categories. Then* F =G.

Proof. By assumption, the underlying functors of the multifunctors F and G are the same and are equal to the functor φ. Let us prove that F f = Gf, for each f : X1, . . . , Xn → Y. The proof is by induction on n. There is nothing to prove if n = 1.

Suppose thatn = 0, i.e.,f is a morphism ()→Y. Then since F and Gare multifunctors, F f =F(u·f) =F u·F f , Gf =G(u·f) =Gu·Gf .

Since F and G coincide on morphisms with one source object, it follows that F f =Gf. Furthermore,

hence F f =Gf. The induction step follows from the commutative diagram C(X2, . . . , Xn;C(X1;Y)) D(φX2, . . . , φXn;φC(X1;Y))

and a similar diagram for G, which are particular cases of Proposition 3.20.

5.12. The bijectivity of U on 2-morphismsThe following proposition implies that U is bijective on 2-morphisms.

5.13. Proposition. *Let* F, G : C → D *be multifunctors between closed multicategories*
*with unit objects. Denote by* Φ = (φ,φ, φˆ ^{0}) *and* Ψ = (ψ,ψ, ψˆ ^{0}) *the corresponding closed*
*functors. Let* r: Φ→Ψ*be a closed natural transformation. Then*r *is also a multinatural*
*transformation* F →G:C→D*.*

Proof.We must prove that, for each f :X1, . . . , Xn→Y, the equation
F f·rY = (rX_{1}, . . . , rXn)·Gf

holds true. The proof is by induction onn. Suppose thatn= 0, and thatf is a morphism ()→Y. The axiom CN1

1 ^{φ}

−→0 F1−^{r}→^{1} G1

=ψ^{0}
implies

()−→^{F u} F1−^{r}→^{1} G1

=Gu.

It follows that

F f ·rY =F u·F f·rY =F u·r_{1}·Gf =Gu·Gf =Gf,

where the second equality is due to the naturality of r. There is nothing to prove in the case n = 1. Suppose that n >1. It suffices to prove that

hF f·rYi=h(rX1, . . . , rXn)·Gfi:F X2, . . . , F Xn→D(F X1;GY).

By Lemma 3.14,(c), the left hand side expands out as hF fi · D(1;rY), which by
for-mula (5.3) is equal to Fhfi ·φˆ·D(1;rY). By Lemma 3.14,(b), the right hand side of the
equation in question is equal to (rX_{2}, . . . , rXn)· hGfi ·D(rX_{1}; 1), which by formula (5.3)
is equal to (rX2, . . . , rXn)·Ghfi ·ψˆ·D(rX1; 1). By the induction hypothesis, the latter is
equal toFhfi ·rC(X1;Y)·ψˆ·D(rX1; 1). The required equation follows then from the axiom
CN2.

5.14. The essential surjectivity of U Let us prove that for each closed categoryV
there is a closed multicategory V with a unit object whose underlying closed category is
isomorphic to V. First of all, notice that by Theorem 2.19 we may (and we shall) assume
in what follows that V is a closed category in the sense of Eilenberg and Kelly; i.e., that
V is equipped with a functor V : V → S such that VV(−,−) = V(−,−) : V^{op} ×V → S
and the axiom CC5’ is satisfied. In particular, we can use the whole theory of closed
categories developed in [2] without any modifications. We are now going to construct a
closed multicategory Vwith a unit object whose underlying closed category is isomorphic
toV. The construction is based on ideas of Laplaza’s paper [9].

We begin by recalling that for each object X of the category V one can assign a
V-functorL^{X} :V→V, and for eachf ∈VV(X, Y) =V(X, Y) there is a uniqueV-natural

transformation L^{f} : L^{Y} → L^{X} : V → V such that (V(L^{f})Y)1Y = f, see Examples 2.13,
2.15, 2.21, or [2, Section 9]. Moreover, by [2, Proposition 9.2] the assignments X 7→L^{X}
and f 7→ L^{f} determine a fully faithful functor from the category V^{op} to the category
V-Cat(V,V) of V-functors V → V and their V-natural transformations. For us it is
more convenient to write it as functor from V to V-Cat(V,V)^{op}. Note that the latter
category is strict monoidal with the tensor product given by composition of V-functors.

More precisely, the tensor product ofF and Gin the given order isF G=F ·G=G◦F.
Consider the multicategory associated withV-Cat(V,V)^{op} (see Example 3.3) and consider
its full submulticategory whose objects areV-functorsL^{X},X ∈ObV. That is, in essence,
our V. More precisely, ObV= ObVand

V(X1, . . . , Xn;Y) =V-Cat(V,V)^{op}(L^{X}^{1} ·. . .·L^{X}^{n}, L^{Y})

=V-Cat(V,V)(L^{Y}, L^{X}^{n}◦ · · · ◦L^{X}^{1}).

Identities and composition coincide with those of the multicategory associated with the
strict monoidal categoryV-Cat(V,V)^{op}. Note that by Proposition 2.20 there is a bijection

Γ :V(X1, . . . , Xn;Y)→(V ◦L^{X}^{n}◦ · · · ◦L^{X}^{1})Y, f 7→(V fY)1Y.

5.15. Theorem. *The multicategory* V *is closed and has a unit object. The underlying*
*closed category of* V *is isomorphic to* V*.*

Proof. First, let us check that the multicategory V is closed. By Proposition 3.9, it
suffices to prove that for each pair of objects X and Z there exist an internal Hom-object
V(X;Z) and an evaluation morphism ev^{V}_{X;Z} :X,V(X;Z)→Z such that the map

ϕ :V(Y1, . . . , Y_{n};V(X;Z))→V(X, Y1, . . . , Y_{n};Z), f 7→(1X, f)·ev^{V}_{X;Z},

is bijective, for each sequence of objects Y_{1}, . . . , Yn. We set V(X;Z) = V(X, Z). The
evaluation map ev^{V}_{X;Z} : X,V(X;Z) → Z is by definition a V-natural transformation
L^{Z} →L^{V}^{(X,Z)}◦L^{X}. We define it by requesting (V(ev^{V}_{X;Z})Z)1Z = 1V(X,Z) (we extensively
use the representation theorem for V-functors in the form of Proposition 2.20). Let us
check that the map ϕ is bijective. Note that the codomain of ϕ identifies via the map Γ
with the set (V ◦L^{Y}^{n} ◦ · · · ◦L^{Y}^{1} ◦L^{X})Z, and that the domain of ϕ identifies via Γ with
the set

(V ◦L^{Y}^{n}◦ · · · ◦L^{Y}^{1})V(X, Z) = (V ◦L^{Y}^{n}◦ · · · ◦L^{Y}^{1} ◦L^{X})Z.

The bijectivity of ϕ follows readily from the diagram

V(Y1, . . . , Yn;V(X;Z)) V(X, Y1, . . . , Yn;Z)

(V ◦L^{Y}^{n}◦ · · · ◦L^{Y}^{1} ◦L^{X})Z

ϕ

Γ Γ

whose commutativity we are going to establish. Take an f ∈V(Y1, . . . , Yn;V(X;Z)), i.e.,

Thus we conclude that Vis a closed multicategory.

Let us check that1∈ObVis a unit object ofV. By definition, a morphismu: ()→1
is a V-natural transformation L^{1} → Id. We let it be equal to i^{−1}, which is a V-natural
transformation by [2, Proposition 8.5]. Then for each object X of V holds

V(u; 1) = (u,1)·ev^{V}_{1;X} :V(1;X)→X,
i.e., V(u; 1) is the V-natural transformation

L^{X} ^{ev}

V

−−−→1;X L^{V}^{(1;X}^{)}◦L^{1} ^{L}

V(1,X)u

−−−−−→L^{V}^{(1}^{,X)}.

We claim that it coincides with L^{i}^{−1}^{X} and hence is invertible. Indeed, applying Γ to the
above composite we obtain

Let us now describe the underlying closed category of the closed multicategory V.
Its objects are those of V, and for each pair of objects X and Y the set of morphisms
from X to Y is V(X;Y) = V-Cat(V,V)(L^{Y}, L^{X}). The unit object is 1 and the internal
Hom-object V(X;Y) coincides with V(X, Y). For each object X, the identity morphism
1^{V}_{X} : () → V(X;X), i.e., a V-natural transformation L^{V}^{(X,X)} → Id, is found from the
Applying Γ to both sides we find that

V((1^{V}_{X}L^{X})◦ev^{V}_{X}_{;X})X

HereV(1^{V}_{X})V(X,X) :V(V(X, X),V(X, X))→V(X, X). The morphismjX of the underlying
closed category of V is a V-natural transformation L^{V}^{(X,X)} → L^{1}; it is found from the
equation

L^{V}^{(X,X)} −→^{j}^{X} L^{1}−→^{u} Id

= 1^{V}_{X}.
Applying Γ to both sides we obtain

V(u◦jX)^{V}(X,X)

1^{V}(X,X) =V(1^{V}_{X})^{V}(X,X)1^{V}(X,X),
i.e.,

V i^{−1}_{V}_{(X,X)}

V(jX)V(X,X)1V(X,X)

= 1X, or equivalently

(V(jX)V(X,X))1V(X,X) = (V iV(X,X))1X =jX,

where the last equality is the axiom CC5’. Therefore, jX = L^{j}^{X} : L^{V}^{(X,X)} → L^{1}. It
also follows by construction that iX for the underlying closed category of the closed
multicategory V is (V(u; 1))^{−1} = (L^{i}^{−1}^{X} )^{−1} =L^{i}^{X}.

Let us compute the morphism L^{X}_{Y Z} : V(Y;Z) → V(V(X;Y);V(X;Z)). First note
that ev^{V}_{X}_{;Y} : X,V(X;Y) → Y is the V-natural transformation L^{Y} → L^{V}^{(X,Y}^{)}◦L^{X} with
components

(ev^{V}_{X}_{;Y})Z =L^{X}_{Y Z} :V(Y, Z)→V(V(X, Y),V(X, Z)).

In other words, ev^{V}_{X;Y} =L^{X}_{Y,−}. Indeed, applying Γ to both side of the equation in question
we obtain an equivalent equation

(V(ev^{V}_{X}_{;Y})Y)1Y = (V L^{X}_{Y Y})1Y.

Since V L^{X} =V(X,−), it follows that (V L^{X}_{Y Y})1Y = 1^{V}(X,Y), so that the above equation is
just the definition of ev^{V}_{X}_{;Y}.

The morphism L^{X}_{Y Z} : V(Y;Z) → V(V(X;Y);V(X;Z)) is uniquely determined by
requesting that the diagram

X,V(X;Y),V(Y;Z) X,V(X;Y),V(V(X;Y);V(X;Z))

X,V(X;Z)

Y,V(Y;Z) Z

1,1,L^{X}_{Y Z}

1,ev^{V}V(X;Y);V(X;Z)

ev^{V}_{X;Z}
ev^{V}_{X;Y},1

ev^{V}_{Y}_{;Z}

in the multicategory V, or equivalently the diagram

L^{Z} L^{V}^{(X,Z)}◦L^{X}

L^{V}^{(}^{V}^{(X,Y}^{),}^{V}^{(X,Z))}◦L^{V}^{(X,Y}^{)}◦L^{X}

L^{V}^{(Y,Z)}◦L^{Y} L^{V}^{(Y,Z)}◦L^{V}^{(X,Y}^{)}◦L^{X}

ev^{V}_{X;Z}

ev^{V}V(X,Y);V(X,Z)L^{X}

L^{X}_{Y Z}L^{V}^{(X,Y}^{)}L^{X}
ev^{V}_{Y;Z}

L^{V}^{(Y,Z)}ev^{V}_{X;Y}

in the category V-Cat(V,V) commute. Applying Γ to both paths in the latter diagram we obtain

V(L^{X}_{Y Z})V(V(X,Y),V(X,Z))

V(ev^{V}V(X,Y);V(X,Z))V(X,Z)

V(ev^{V}_{X;Z})Z

1Z

=V(V(Y, Z),(ev^{V}_{X;Y})Z)(V(ev^{V}_{Y}_{;Z})Z)1Z,
or equivalently

V(L^{X}_{Y Z})V(V(X,Y),V(X,Z))

1V(V(X,Y),V(X,Z)) = (ev^{V}_{X;Y})Z =L^{X}_{Y Z}.
In other words, L^{X}_{Y Z} for the underlying closed category of V isL^{L}^{X}^{Y Z}.

Let us denote the underlying closed category of the multicategory V by the same
symbol. There is a closed functor (L,1,1) : V → V, where L : V → V is given by
X 7→X, f 7→L^{f}, and the morphisms V(X, Y)→V(X;Y) and 1→1 are the identities.

The axioms CF1–CF3 follow readily from the above description of the closed category V. Clearly, the functor Lis an isomorphism. The theorem is proven.

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*Department of Computer Science*

*National University of Ireland Maynooth*
*Maynooth, Ireland*

Email: manzyuk@gmail.com

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